Quantum Galilei group as quantum reference frame transformations

TL;DR

The paper addresses how quantum reference frame (QRF) transformations relate to quantum symmetries by establishing a correspondence between QRF transformations and a quantum deformation of the Galilei group with commuting time, analyzed via a phase-space representation. It identifies a unique quantum Galilei group compatible with QRF structure and shows that, at first order in the deformation parameter, the quantum-group generators reproduce the QRF dynamical algebra when the parameter is identified with the inverse mass of the QRF. The work provides explicit phase-space realizations and mappings between QRF operators and quantum-group coordinates, bridging two formalisms and revealing that QRF transformations on superpositions of semiclassical states are physically meaningful within this quantum-group framework. It also connects to the Poisson–Lie limit describing dynamical classical frames and conjectures an all-order quantum Galilei group description for general QRF states, suggesting future extensions to higher dimensions and relativistic settings.

Abstract

Quantum groups have been widely explored as a tool to encode possible nontrivial generalisations of reference frame transformations, relevant in quantum gravity. In quantum information, it was found that the reference frames can be associated to quantum particles, leading to quantum reference frames transformations. The connection between these two frameworks is still unexplored, but if clarified it will lead to a more profound understanding of symmetries in quantum mechanics and quantum gravity. Here, we establish a correspondence between quantum reference frame transformations and transformations generated by a quantum deformation of the Galilei group with commutative time, taken at first order in the quantum deformation parameter. This is found once the quantum group noncommutative transformation parameters are represented on the phase space of a quantum particle, and upon setting the quantum deformation parameter to be proportional to the inverse of the mass of the particle serving as the quantum reference frame. These results allow us to show that quantum reference frame transformations are physically relevant when the state of the quantum reference frame is in a quantum superposition of semiclassical states. We conjecture that the all-order quantum Galilei group describes quantum reference frame transformations between more general quantum states of the quantum reference frame.